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Chapter 12: Problem 56

To what temperature, in degrees Celsius, must a 25.5 -mL. sample of oxygen at \(90^{\circ} \mathrm{C}\) be cooled for its volume to decrease to 21.5 mL? Assume the pressure and mass of the gas are constant.

Short Answer

Expert verified
The final temperature is approximately \(33.2 ^{\circ}C\).

Step by step solution

01

Identify the Known Quantities

We know the initial volume \(V_1 = 25.5\) mL, the initial temperature \(T_1 = 90^{\circ}C\), which is equivalent to \(90 + 273.15 = 363.15\ K\), and the final volume \(V_2 = 21.5\) mL. We need to find the final temperature \(T_2\) in degrees Celsius.
02

Apply Charles's Law

Charles's Law relates temperature and volume for a gas at constant pressure: \(\frac{V_1}{T_1} = \frac{V_2}{T_2}\). Using Charles's Law, we can solve for \(T_2\).
03

Substitute the Known Values into Charles's Law

We have \(\frac{25.5}{363.15} = \frac{21.5}{T_2}\). Solving for \(T_2\), multiply both sides by \(T_2\) and then divide both sides by \(\frac{25.5}{363.15}\) to isolate \(T_2\).
04

Calculate the Final Temperature in Kelvin

Rearranging and calculating gives: \[ T_2 = \left( \frac{21.5 \times 363.15}{25.5} \right) \approx 306.35 \ K. \]
05

Convert Temperature from Kelvin to Celsius

Convert the final temperature from Kelvin to Celsius by subtracting 273.15: \[ T_2 = 306.35 - 273.15 \approx 33.2 \ ^{\circ}C. \]

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Key Concepts

These are the key concepts you need to understand to accurately answer the question.

Gas Laws
Gas laws are a fundamental part of understanding how gases behave under different conditions. They provide the relationships between variables such as volume, temperature, and pressure.
  • Boyle's Law states that for a fixed amount of gas at constant temperature, the volume is inversely proportional to the pressure.
  • Charles's Law indicates that for a fixed amount of gas at constant pressure, the volume is directly proportional to the temperature, as stated by \( \frac{V_1}{T_1} = \frac{V_2}{T_2} \). This means that when the temperature of a gas increases, its volume increases as well.
Understanding these laws is crucial for solving problems involving changes in gas conditions, as they help predict how a gas sample's volume and temperature will change when the conditions are altered.
Temperature Conversion
Temperature conversion is essential when working with gas laws because temperatures need to be in Kelvin. This is because Kelvin is an absolute scale starting at absolute zero, where molecular motion stops completely. Here's how you convert:
  • To convert Celsius to Kelvin, add 273.15 to the Celsius temperature.
  • To convert Kelvin back to Celsius, subtract 273.15 from the Kelvin temperature.
For example, if you have a temperature of \(90^{\circ}C\), converting it to Kelvin would be \(90 + 273.15 = 363.15\ K\). This conversion is crucial when using Charles's Law, as the laws require temperatures to be in Kelvin for accuracy.
Volume and Temperature Relationship
The volume and temperature relationship for gases is wonderfully explained by Charles's Law, which is ideal in scenarios of maintained pressure.In simple terms, if the temperature of a gas is increased, the particles inside gain energy, moving more rapidly, thereby increasing the volume if the container allows. Conversely, as the temperature cools, particles calm down, reducing the volume. When solving for a decreased volume due to cooling, as in the sample problem, we use this relationship:\[ \frac{V_1}{T_1} = \frac{V_2}{T_2} \]Here, if you know the initial volume and temperature and need to find the final temperature after a volume change, rearrange the formula to:\[ T_2 = \frac{V_2 \times T_1}{V_1} \]This clear relationship helps us understand how energy changes (in temperature) directly affect the space taken up by gas molecules, illustrating the dynamic nature of gases.

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Most popular questions from this chapter

Iron carbonyl can be made by the direct reaction of iron metal and carbon monoxide. $$ \mathrm{Fe}(\mathrm{s})+5 \mathrm{CO}(\mathrm{g}) \longrightarrow \mathrm{Fe}(\mathrm{CO})_{5}(\ell) $$ What is the theoretical yield of \(\mathrm{Fe}(\mathrm{CO})_{5}\), if \(3.52 \mathrm{g}\) of iron is treated with CO gas having a pressure of \(732 \mathrm{mm}\) Hg in a \(5.50-\) I. Hask at \(23^{\circ} \mathrm{C} ?\)

One of the cylinders of an automobile engine has a volume of \(400 . \mathrm{cm}^{3} .\) The engine takes in air at a pressure of 1.00 atm and a temperature of \(15^{\circ} \mathrm{C}\) and compresses the air to a volume of \(50.0 \mathrm{cm}^{3}\) at \(77^{\circ} \mathrm{C}\). What is the final pressure of the gas in the cylinder? (The ratio of before and after volumes - in this case, 400: 50 or \(8: 1-\) is called the compression ratio.

In the text it is stated that the pressure of 8.00 mol of \(\mathrm{Cl}_{2}\) in a \(4.00-\mathrm{L}\) tank at \(27.0^{\circ} \mathrm{C}\) should be 29.5 atm if calculated using the van der Waals's equation. Verify this result and compare it with the pressure predicted by the ideal gas law.

The pressure of a gas is \(440 \mathrm{mm}\) Hg. Express this pressure in units of (a) atmospheres, (b) bars, and (c) kilopascals.

If you place \(2.25 \mathrm{g}\) of solid silicon in a \(6.56-\mathrm{L}\). flask that contains \(\mathrm{CH}_{3} \mathrm{Cl}\) with a pressure of \(585 \mathrm{mm} \mathrm{Hg}\) at \(25^{\circ} \mathrm{C}\) what mass of dimethyldichlorosilane, \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{SiCl}_{2}(\mathrm{g}),\) can be formed? $$\mathrm{Si}(\mathrm{s})+2 \mathrm{CH}_{3} \mathrm{Cl}(\mathrm{g}) \longrightarrow\left(\mathrm{CH}_{3}\right)_{2} \mathrm{Si} \mathrm{Cl}_{2}(\mathrm{g})$$ What pressure of \(\left(\mathrm{CH}_{3}\right)_{2} \mathrm{SiCl}_{2}(\mathrm{g})\) would you expect in this same flask at \(95^{\circ}\) C on completion of the reaction? (Dimethyldichlorosilane is one starting material used to make silicones, polymeric substances used as lubricants, antistick agents, and water-proofing caulk.)
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